Existence and Iteration of Positive Solutions to Third-Order BVP for a Class of p

We investigate the existence and iteration of positive solutions for the following third-order p-Laplacian dynamic equations on time scales: (ϕp(uΔΔ(t)))∇+q(t)f(t,u(t),uΔΔ(t))=0,  t∈[a,b],αu(ρ(a))-βuΔ(ρ(a))=0,  γu(b)+δuΔ(b)=0,  uΔΔ(ρ(a))=0, where ϕp(s) is p-Laplacian operator; that is, ϕp(s)=sp-2s,  p>1,  ϕp-1=ϕq, and 1/p+1/q=1. By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions

Positive Solutions for Third-Order p

We study the following third-order p-Laplacian functional dynamic equation on time scales: Φp(uΔ∇(t))∇+a(t)f(u(t),u(μ(t)))=0, t∈0,TT,  u(t)=φ(t),  t∈-r,0T,  uΔ(0)=uΔ∇(T)=0, and u(T)+B0(uΔ(η))=0. By applying the Five-Functional Fixed Point Theorem, the existence criteria of three positive solutions are established

Some New Existence Results of Positive Solutions to an Even-Order Boundary Value Problem on Time Scales

We consider a high-order three-point boundary value problem. Firstly, some new existence results of at least one positive solution for a noneigenvalue problem and an eigenvalue problem are established. Our approach is based on the application of three different fixed point theorems, which have extended and improved the famous Guo-Krasnosel'skii fixed point theorem at different aspects. Secondly, some examples are included to illustrate our results

Existence of positive solutions for multi-point time scale boundary value problems on infinite intervals

In this paper, we establish the criteria for the existence of at least one and three positive solutions for a nonlinear second order multi-point time scale boundary value problem on infinite interval based on the Leray- Schauder fixed point theorem and the five functional fixed point theorem, respectively. © 2017 Mathematical Research Press. All rights reserved

Existence of positive solutions for third-order semipositone boundary value problems on time scales

In this paper, we consider the existence of positive solutions for a semipositone third-order nonlinear ordinary differential equation on time scales. In suitable growth conditions, by considering the properties on time scales and establishing a special cone, some new results on the existence of positive solutions are established when the nonlinearity is semipositone

Steady advection-diffusion around finite absorbers in two-dimensional potential flows

We perform an exhaustive study of the simplest, nontrivial problem in advection-diffusion -- a finite absorber of arbitrary cross section in a steady two-dimensional potential flow of concentrated fluid. This classical problem has been studied extensively in the theory of solidification from a flowing melt, and it also arises in Advection-Diffusion-Limited Aggregation. In both cases, the fundamental object is the flux to a circular disk, obtained by conformal mapping from more complicated shapes. We construct the first accurate numerical solution using an efficient new method, which involves mapping to the interior of the disk and using a spectral method in polar coordinates. Our method also combines exact asymptotics and an adaptive mesh to handle boundary layers. Starting from a well-known integral equation in streamline coordinates, we also derive new, high-order asymptotic expansions for high and low P\'eclet numbers (\Pe). Remarkably, the `high' \Pe expansion remains accurate even for such low \Pe as $10^{-3}$. The two expansions overlap well near \Pe = 0.1, allowing the construction of an analytical connection formula that is uniformly accurate for all \Pe and angles on the disk with a maximum relative error of 1.75%. We also obtain an analytical formula for the Nusselt number ($\Nu$) as a function of the P\'eclet number with a maximum relative error of 0.53% for all possible geometries. Because our finite-plate problem can be conformally mapped to other geometries, the general problem of two-dimensional advection-diffusion past an arbitrary finite absorber in a potential flow can be considered effectively solved.Comment: 29 pages, 12 figs (mostly in color